Global small solutions to the compressible 2D magnetohydrodynamic system without magnetic diffusion

TL;DR

This paper proves the global existence, uniqueness, and decay rates of smooth solutions for the 2D compressible magnetohydrodynamic system without magnetic diffusion, using a novel diagonalization approach and advanced Fourier analysis.

Contribution

It introduces a new diagonalization method for coupled linearized equations, enabling analysis of the system without magnetic diffusion and deriving explicit decay rates.

Findings

Global smooth solutions exist and are unique near equilibrium.

Explicit large-time decay rates for Sobolev norms are established.

The new diagonalization approach simplifies analysis of complex coupled systems.

Abstract

This paper establishes the global existence and uniqueness of smooth solutions to the two-dimensional compressible magnetohydrodynamic system when the initial data is close to an equilibrium state. In addition, explicit large-time decay rates for various Sobolev norms of the solutions are also obtained. These results are achieved through a new approach of diagonalizing a system of coupled linearized equations. The standard method of diagonalization via the eigenvalues and eigenvectors of the matrix symbol is very difficult to implement here. This new process allows us to obtain an integral representation of the full system through explicit kernels. In addition, in order to overcome various difficulties such as the anisotropicity and criticality, we fully exploit the structure of the integral representation and employ extremely delicate Fourier analysis and associated estimates.